A path-addition adds an internally vertex-disjoint path \(P\) 
between any two vertices of a graph. Such paths are also known as
ears. They are used in ear decompositions of 2-connected graphs
[1]} and in subdivisions. We wish to use
path-additions so that they preserve a given class of graphs.
Therefore, the added paths are long. Their length is at least \(cn\) 
for some \(c > 0\)  and \(n\) -vertex graphs. This property distinguishes
our path-addition operation from ear decompositions
[1]}. In comparison with subdivision, a
path-addition adds a (long) path between any two
vertices, whereas a subdivision can do so if there is an edge. Hence,
path-additions can create a subdivision of \(K_n\)  from any non-empty graph. A
more general version of path-addition was introduced in
[3]} and studied in [4]}. For a graph \(G\) 
and a path \(P\) , let \(G \oplus P\)  denote the graph obtained by adding
the vertices and edges of \(P\)  to \(G\) , where the internal vertices of
\(P\)  are new and have degree two. A class of graphs \(\mathcal {G}\)  is
closed under path-addition if there is some function
\(D_{\mathcal {G}}(n)\)  so that \(G \oplus P\)  is in \(\mathcal {G}\)  if \(G\) 
is an \(n\) -vertex graph in \(\mathcal {G}\)  and \(P\)  has length at least
\(D_{\mathcal {G}}(n)\) .
